I'm not a maths guy, or a physics guy (except in a vary amateur sense,) but I'll have a go at this. It doesn't need a great deal of maths to grasp the priciple; just basic adding and subtracting of a few simple numbers.
Say the speed of light is absolute. This means that two observers, measuring the speed of the same photon, will get the same result.
This seems obvious, till you realise that each observer is measuring the photon's speed relative to herself. (or himself.)
Now. take a simple analogy. You and I are travelling along a wide straight road, in our cars, in the same direction. And we both have radar 'speed measuring devices' in our cars.
Say at one point you are a mile or so ahead of me. Your speedo says you're doing sixty, and my speedo says I'm doing seventy. If you turn your radar device onto my car, you'll read that I'm coming up behind you at ten miles an hour, and my radar device will tell me you are coming back towards me at ten miles an hour.
OK, nothing weird so far.
So suppose we both train our devices on the bridge up ahead. If your speedo tells you your car is moving at sixty, your radar will read the bridge coming back at you at sixty. (Because the bridge is standing still on the road.) And since I'm moving at seventy over the road, my radar reads the bridge as coming back toward me at seventy.
Your radar says the bridge is 'doing sixty', and my radar measures the same bridge as 'doing seventy'. We don't think this is odd, because we know we are moving at different speeds relative to each other. The ten mph difference in our relative speeds is reflected precisely in the ten mph difference we get when we both measure the same object.
And this works just the same if the object is itself moving.
Suppose an ambulance comes tearing up behind us. Let's say the ambulance speedo says it's doing ninety. We both train our radars on the ambulance. Your radar tells you the ambulance is closing on you at thirty mph. (Its ninety minus your sixty.) My radar tells me its closing on me at twenty. (Its ninety minus my seventy)
Again, the difference in our observations is exactly the same as the difference we measure in our own speeds.
(And it doesn't matter if we're moving in opposite directions either. We just use plusses instead of minuses.)
The important point is that, if we can measure a difference in our own speeds relative to each other, then we will measure a difference of the same size in the speed of anything that we both measure.
This is how it turns out for all ordinary speeds.
But realising that the speed of light is the same for all observers, (to see exactly how he realised this would take quite a bit of maths; more than I've got, so I'll skip it.) Einstein showed that the time and space must themselves be different for the two observers.
This is because 'speed' is only 'distance' divided by 'time'. Miles per hour, metres per second, inches per century, they're all just lengths of space divided by lengths of time. So if the two observers are moving at vastly different speeds relative to the nearest planet, and they each measure the speed (relative to themselves) of a passing photon, and they both get the same result, then there must be a discrepancy in the way the observers are measuring time and space. And if all their measuring instruments are accurate, then this discrepancy must be real, and it must be related to the fact that they are moving relative to each other. Because there is no other difference between them, or between their measuring devices or techniques.
The relativity of space and time simply means that two observers, moving at some speed relative to each other, will measure time and space differently.
In other words, the absolute value of the speed of light doesn't contradict the relativity of time and space; on the exact contrary, it is the absolute speed of light which proves it.
Thanks -- I do better with examples, such as you've provided, than with abstractions. I'm chewing on the matter now, using simple paper and pen sketches. rebecca